Midpoints for Thompson's metric on symmetric cones

We characterise the affine span of the midpoints sets, \(M(x,y)\), for Thompson's metric on symmetric cones in terms of a translation of the zero-component of the Peirce decomposition of an idempotent. As a consequence we derive an explicit formula for the dimension of the affine span of \(M(x,y)\) in case the associated Euclidean Jordan algebra is simple. In particular, we find for \(A\) and \(B\) in the cone positive definite Hermitian matrices that \(dim(aff M(A,B)) = q^2\), where \(q\) is the number of eigenvalues \(\mu\) of \(A^{-1}B\), counting multiplicities, such that \(\mu ≠ max\{\lambda_+(A^{-1}B),\lambda_-(A^{-1}B)^{-1}\},\) where \(\lambda_+(A^{-1}B) := max\{\lambda:\lambda \in \sigma(A^{-1}B)\}\) and \(\lambda_-(A^{-1}B) := min\{\lambda:\lambda \in \sigma(A^{-1}B)\}\). These results extend work by Y. Lim [18].